Antisymmetry of solutions for some weighted elliptic problems

TL;DR

This paper investigates the antisymmetry, uniqueness, and monotonicity of solutions to weighted elliptic problems, introducing a new rearrangement technique and establishing conditions under which solutions are antisymmetric.

Contribution

It introduces a continuous odd rearrangement for one-dimensional solutions and improves conditions for antisymmetry from convexity to monotonicity of weights.

Findings

Rearrangement decreases energy functional under convexity conditions.

Solutions are antisymmetric or odd when weights satisfy certain conditions.

Counterexamples show antisymmetry does not always hold.

Abstract

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.